Higher-rank $F_K$ conjecture

Let $G$ be a root system of rank $r$, with Weyl group $W$, positive roots $\Delta^+$, weight lattice $P$, root lattice $Q$, Weyl vector $\rho$, and dominant weights $P_+$. Let $\mathbf{x}=(x_1,\dots,x_r)$, and write $x^\alpha$ for the monomial associated to a root $\alpha$. Higher-rank $F_K$ conjecture. For every knot $K$, there exists a series

$$F_K^G(\mathbf{x},q)=\frac{1}{|W|}\sum_{\beta\in P_+\cap(Q+\rho)}f_\beta^G(q)\sum_{w\in W}(-1)^{l(w)}x^{w(\beta)},$$

with Laurent-series coefficients $f_\beta^G(q)$ having integer coefficients, whose asymptotic expansion is

$$F_K^G(\mathbf{x},\mathrm{e}^{\hbar})=\prod_{\alpha\in\Delta^+}(x^{\alpha/2}-x^{-\alpha/2})\sum_{j\geq 0}\frac{P_j(\mathbf{x})}{\left(\prod_{\alpha\in\Delta^+}\Delta_K(x^\alpha)\right)^{2j+1}}\frac{\hbar^j}{j!},$$

where $P_j(\mathbf{x})\in\mathbb{Z}[x_1,x_1^{-1},\dots,x_r,x_r^{-1}]$ and $P_0=1$, with semiclassical limit

$$\lim_{q\to 1}F_K^G(\mathbf{x},q)=\prod_{\alpha\in\Delta^+}\frac{x^{\alpha/2}-x^{-\alpha/2}}{\Delta_K(x^\alpha)},$$

and it is annihilated by the higher-rank quantum $A$-polynomial:

$$\hat{A}_K(\hat{x}_1,\hat{y}_1,\dots,\hat{x}_r,\hat{y}_r)F_K^G(\mathbf{x},q)=0.$$

This is the proposed root-system generalization of the $G={\rm SU}(2)$ invariant. The construction is supported by computations, including torus knots and the figure-eight knot, but remains conjectural in general.